NAG Library Function Document

nag_deviates_gamma_dist (g01ffc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_deviates_gamma_dist (g01ffc) returns the deviate associated with the given lower tail probability of the gamma distribution.

2 Specification

#include <nag.h>

#include <nagg01.h>

double

nag_deviates_gamma_dist (double p, double a, double b, double tol, NagError *fail)

3 Description

The deviate,

g_{p}

, associated with the lower tail probability,

p

, of the gamma distribution with shape parameter

α

and scale parameter

β

, is defined as the solution to

P (G \leq g_{p} : α, β) = p = \frac{1}{β^{α} Γ (α)} \int_{0}^{g_{p}} e^{- G / β} G^{α - 1} d G, 0 \leq g_{p} < \infty; α, β > 0 .

The method used is described by Best and Roberts (1975) making use of the relationship between the gamma distribution and the

χ^{2}

-distribution.

Let

y = 2 \frac{g_{p}}{β}

. The required

y

is found from the Taylor series expansion

y = y_{0} + \sum_{r} \frac{C_{r} (y_{0})}{r!} {(\frac{E}{ϕ (y_{0})})}^{r},

where

y_{0}

is a starting approximation

$C_{1} (u) = 1$ ,
$C_{r + 1} (u) = (r Ψ + \frac{d}{d u}) C_{r} (u)$ ,
$Ψ = \frac{1}{2} - \frac{α - 1}{u}$ ,
$E = p - \int_{0}^{y_{0}} ϕ (u) d u$ ,
$ϕ (u) = \frac{1}{2^{α} Γ (α)} e^{- u / 2} u^{α - 1}$ .

For most values of

p

and

α

the starting value

y_{01} = 2 α {(z \sqrt{\frac{1}{9 α}} + 1 - \frac{1}{9 α})}^{3}

is used, where

z

is the deviate associated with a lower tail probability of

p

for the standard Normal distribution.

For

p

close to zero,

y_{02} = {(p α 2^{α} Γ (α))}^{1 / α}

is used.

For large

p

values, when

y_{01} > 4.4 α + 6.0

y_{03} = - 2 [\ln (1 - p) - (α - 1) \ln (\frac{1}{2} y_{01}) + \ln (Γ (α))]

is found to be a better starting value than

y_{01}

For small

α

(α \leq 0.16)

p

is expressed in terms of an approximation to the exponential integral and

y_{04}

is found by Newton–Raphson iterations.

Seven terms of the Taylor series are used to refine the starting approximation, repeating the process if necessary until the required accuracy is obtained.

4 References

Best D J and Roberts D E (1975) Algorithm AS 91. The percentage points of the

χ^{2}

distribution Appl. Statist. 24 385–388

5 Arguments

1: p – doubleInput: On entry: $p$ , the lower tail probability from the required gamma distribution.
Constraint: $0.0 \leq p < 1.0$ .
2: a – doubleInput: On entry: $α$ , the shape parameter of the gamma distribution.
Constraint: $0.0 < a \leq 10^{6}$ .
3: b – doubleInput: On entry: $β$ , the scale parameter of the gamma distribution.
Constraint: $b > 0.0$ .
4: tol – doubleInput: On entry: the relative accuracy required by you in the results. The smallest recommended value is $50 \times δ$ , where $δ = \max (10^{- 18}, machine precision)$ . If nag_deviates_gamma_dist (g01ffc) is entered with tol less than $50 \times δ$ or greater or equal to $1.0$ , then $50 \times δ$ is used instead.
5: fail – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

On any of the error conditions listed below, except $fail . code =$ NE_ALG_NOT_CONV, nag_deviates_gamma_dist (g01ffc) returns $0.0$ .
NE_ALG_NOT_CONV: The algorithm has failed to converge in 100 iterations. A larger value of tol should be tried. The result may be a reasonable approximation.
NE_GAM_NOT_CONV: The series used to calculate the gamma function has failed to converge. This is an unlikely error exit.
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_PROBAB_CLOSE_TO_TAIL: The probability is too close to $0.0$ for the given a to enable the result to be calculated.
NE_REAL_ARG_GE: On entry, $p = ⟨value⟩$ .
Constraint: $p < 1.0$ .
NE_REAL_ARG_GT: On entry, $a = ⟨value⟩$ .
Constraint: $a \leq 10^{6}$ .
NE_REAL_ARG_LE: On entry, $a = ⟨value⟩$ .
Constraint: $a > 0.0$ .
On entry, $b = ⟨value⟩$ .
Constraint: $b > 0.0$ .
NE_REAL_ARG_LT: On entry, $p = ⟨value⟩$ .
Constraint: $p \geq 0.0$ .

7 Accuracy

In most cases the relative accuracy of the results should be as specified by tol. However, for very small values of

α

or very small values of

p

there may be some loss of accuracy.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

This example reads lower tail probabilities for several gamma distributions, and calculates and prints the corresponding deviates until the end of data is reached.

NAG Library Function Documentnag_deviates_gamma_dist (g01ffc)